Simplify and expand the following expression: $ \dfrac{4}{x - 5}+\dfrac{5x - 6}{x + 10} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(x - 5)(x + 10)$ Multiply the first term by $\dfrac{x + 10}{x + 10}$ $ \begin{align*} \dfrac{4}{x - 5} \times \dfrac{x + 10}{x + 10} & = \dfrac{(4)(x + 10)}{(x - 5)(x + 10)} \\ & = \dfrac{4x + 40}{(x - 5)(x + 10)}\end{align*} $ Multiply the second term by $\dfrac{x - 5}{x - 5}$ $ \begin{align*} \dfrac{5x - 6}{x + 10} \times \dfrac{x - 5}{x - 5} & = \dfrac{(5x - 6)(x - 5)}{(x + 10)(x - 5)} \\ & = \dfrac{5x^2 - 31x + 30}{(x + 10)(x - 5)}\end{align*} $ Now we have: $ = \dfrac{4x + 40}{(x - 5)(x + 10)} + \dfrac{5x^2 - 31x + 30}{(x + 10)(x - 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4x + 40 + 5x^2 - 31x + 30}{(x - 5)(x + 10)} $ $ = \dfrac{-27x + 70 + 5x^2}{(x - 5)(x + 10)}$ Expand the denominator: $ = \dfrac{-27x + 70 + 5x^2}{x^2 + 5x - 50}$